Group #22581127 — subgroups

Subgroups — #22581127

Order 8896.

146 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17984889	C₂	1	1	yes	maximal, characteristic, complemented
2	#17984870	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984867	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984830	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984770	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984760	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#14555785	—	2	1	no	—
9	#14555785	C₂²	1	1	yes	characteristic
10	#14555784	—	2	2	no	—
11	#14555784	—	2	1	no	—
12	C₂×Dic₂₇₈	—	2	1	no	—
13	C₂×Dic₂₇₈	C₂²	1	1	yes	characteristic
14	Q₈×D₁₃₉	—	2	1	no	—
15	C₂×D₅₅₆	C₂²	1	1	yes	characteristic
16	#14555774	—	2	1	no	—
17	#14555772	—	2	1	no	—
18	D₁₁₁₂	—	2	1	no	—
19	#14555770	—	2	1	no	—
20	#14555769	C₂²	1	1	yes	characteristic
21	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
22	#41×C₁₃₉	C₂²	1	1	yes	characteristic
23	M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
24	C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
25	C₂²×D₁₃₉	—	2	1	no	—
26	Dic₂₇₈	—	2	2	no	—
27	Dic₂₇₈	—	2	1	no	—
28	Dic₂₇₈	—	2	1	no	—
29	Dic₂₇₈	D₄	1	1	yes	characteristic
30	D₅₅₆	—	2	1	no	—
31	D₅₅₆	—	2	1	no	—
32	D₅₅₆	D₄	1	1	yes	characteristic
33	C₂²⋊D₁₃₉	—	2	2	no	—
34	C₂²⋊D₁₃₉	—	2	1	no	—
35	C₂×Dic₁₃₉	—	2	2	no	—
36	C₂×Dic₁₃₉	D₄	1	2	yes	—
37	C₄×D₁₃₉	—	2	2	no	—
38	C₄×D₁₃₉	—	2	1	no	—
39	C₁₃₉⋊C₈	—	2	1	no	—
40	Q₈×C₁₃₉	D₄	1	1	yes	characteristic
41	D₄×C₁₃₉	—	2	1	no	—
42	D₄×C₁₃₉	D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃₉	—	2	1	no	—
44	C₂×C₄×C₁₃₉	—	2	1	no	—
45	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
46	C₈×C₁₃₉	—	2	1	no	—
47	C₂²×C₁₃₉	—	2	1	no	—
48	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
49	C₂×D₁₃₉	—	2	1	no	—
50	C₂×D₁₃₉	—	4	2	no	—
51	C₂×D₁₃₉	—	4	1	no	—
52	C₄×C₁₃₉	—	2	1	no	—
53	C₄×C₁₃₉	—	4	1	no	—
54	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
55	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
56	Dic₁₃₉	—	2	1	no	—
57	Dic₁₃₉	—	2	2	no	—
58	C₂×C₁₃₉	—	2	1	no	—
59	C₂×C₁₃₉	—	4	1	no	—
60	C₂×C₁₃₉	#120	1	1	yes	characteristic
61	D₁₃₉	—	4	1	no	—
62	D₁₃₉	—	8	1	no	—
63	C₁₃₉	#454	1	1	yes	characteristic, complemented, Sylow
64	#454	—	139	1	no	maximal, Sylow
65	2₋¹⁺⁴	—	139	1	no	—
66	AGL(1,ℤ/8)	—	139	1	no	—
67	AGL(1,ℤ/8)	—	139	1	no	—
68	#124	—	139	1	no	—
69	#104	—	139	1	no	—
70	#104	D₁₃₉	1	1	yes	characteristic, complemented
71	#101	—	139	1	no	—
72	#41	—	278	1	no	—
73	#41	—	278	1	no	—
74	#41	—	278	2	no	—
75	#41	—	139	1	no	—
76	#41	C₂×D₁₃₉	1	1	yes	characteristic
77	C₂×Q₈	—	278	1	no	—
78	C₂×Q₈	—	278	1	no	—
79	C₂×Q₈	—	139	1	no	—
80	C₂×D₄	—	139	1	no	—
81	SD₁₆	—	278	1	no	—
82	SD₁₆	—	278	1	no	—
83	D₈	—	278	1	no	—
84	D₈	—	278	1	no	—
85	M₁₆	—	139	1	no	—
86	M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
87	C₂²⋊C₄	—	278	2	no	—
88	C₄²	C₂×D₁₃₉	1	1	yes	characteristic
89	C₂³	—	278	1	no	—
90	Q₈	—	278	2	no	—
91	Q₈	—	278	1	no	—
92	Q₈	—	278	1	no	—
93	Q₈	—	139	1	no	—
94	Q₈	D₅₅₆	1	1	yes	characteristic
95	D₄	—	2	1	no	—
96	D₄	—	278	2	no	—
97	D₄	—	278	1	no	—
98	D₄	—	278	1	no	—
99	D₄	—	278	1	no	—
100	D₄	—	139	1	no	—
101	D₄	D₅₅₆	1	1	yes	characteristic
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	278	2	no	—
105	C₂×C₄	—	278	2	no	—
106	C₂×C₄	—	278	1	no	—
107	C₂×C₄	—	139	2	no	—
108	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
109	C₈	—	2	1	no	—
110	C₈	—	278	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	278	1	no	—
113	C₂²	—	556	2	no	—
114	C₂²	—	556	1	no	—
115	C₂²	D₄×D₁₃₉	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	4	1	no	—
118	C₄	—	278	1	no	—
119	C₄	—	278	2	no	—
120	C₄	D₄×D₁₃₉	1	1	yes	characteristic
121	C₄	C₂×D₅₅₆	1	1	yes	characteristic
122	C₂	—	2	1	no	—
123	C₂	—	4	1	no	—
124	C₂	—	556	1	no	—
125	C₂	—	1112	1	no	—
126	C₂	#17984812	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984889

C₂

yes

maximal, characteristic, complemented

#17984870

C₂

yes

maximal, characteristic, complemented

#17984867

C₂

yes

maximal, characteristic, complemented

#17984830

C₂

yes

maximal, characteristic, complemented

#17984770

C₂

yes

maximal, characteristic, complemented

#17984760

C₂

yes

maximal, characteristic, complemented

#104×C₁₃₉

C₂

yes

maximal, characteristic, complemented

#14555785

—

#14555785

C₂²

yes

characteristic

#14555784

—

#14555784

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

C₂²

yes

characteristic

Q₈×D₁₃₉

—

C₂×D₅₅₆

C₂²

yes

characteristic

#14555774

—

#14555772

—

D₁₁₁₂

—

#14555770

—

#14555769

C₂²

yes

characteristic

(C₂×D₁₃₉)⋊C₄

—

#41×C₁₃₉

C₂²

yes

characteristic

M₁₆×C₁₃₉

C₂²

yes

characteristic

C₄²×C₁₃₉

C₂²

yes

characteristic

C₂²×D₁₃₉

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

D₄

yes

characteristic

D₅₅₆

—

D₅₅₆

—

D₅₅₆

D₄

yes

characteristic

C₂²⋊D₁₃₉

—

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

D₄

yes

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

Q₈×C₁₃₉

D₄

yes

characteristic

D₄×C₁₃₉

—

D₄×C₁₃₉

D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×D₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₄×C₁₃₉

C₂×D₄

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#120

yes

characteristic

D₁₃₉

—

D₁₃₉

—

C₁₃₉

#454

yes

characteristic, complemented, Sylow

#454

—

139

maximal, Sylow

2₋¹⁺⁴

—

139

—

AGL(1,ℤ/8)

—

139

—

AGL(1,ℤ/8)

—

139

—

#124

—

139

—

#104

—

139

—

#104

D₁₃₉

yes

characteristic, complemented

#101

—

139

—

#41

—

278

—

#41

—

278

—

#41

—

278

—

#41

—

139

—

#41

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

139

—

C₂×D₄

—

139

—

SD₁₆

—

278

—

SD₁₆

—

278

—

D₈

—

278

—

D₈

—

278

—

M₁₆

—

139

—

M₁₆

C₂×D₁₃₉

yes

characteristic

C₂²⋊C₄

—

278

—

C₄²

C₂×D₁₃₉

yes

characteristic

C₂³

—

278

—

Q₈

—

278

—

Q₈

—

278

—

Q₈

—

278

—

Q₈

—

139

—

Q₈

D₅₅₆

yes

characteristic

D₄

—

D₄

—

278

—

D₄

—

278

—

D₄

—

278

—

D₄

—

278

—

100

D₄

—

139

—

101

D₄

D₅₅₆

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

278

—

105

C₂×C₄

—

278

—

106

C₂×C₄

—

278

—

107

C₂×C₄

—

139

—

108

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

109

C₈

—

110

C₈

—

278

—

111

C₂²

—

112

C₂²

—

278

—

113

C₂²

—

556

—

114

C₂²

—

556

—

115

C₂²

D₄×D₁₃₉

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

278

—

119

C₄

—

278

—

120

C₄

D₄×D₁₃₉

yes

characteristic

121

C₄

C₂×D₅₅₆

yes

characteristic

122

C₂

—

123

C₂

—

124

C₂

—

556

—

125

C₂

—

1112

—

126

C₂

#17984812

yes

characteristic

	subgroup	G/H
center	C₂	#17984812
commutator	C₂×C₄×C₁₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₉
Fitting	#104×C₁₃₉	C₂
socle	C₂×C₁₃₉	#120
hypercenter	#104	D₁₃₉
hypocenter	C₁₃₉	#454
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

subgroup

G/H

center

C₂

#17984812

commutator

C₂×C₄×C₁₃₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₃₉

Fitting

#104×C₁₃₉

C₂

C₂×C₁₃₉

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

N	H ≅ G/N	type
1. #17984889	C₂	semidirect
2. #17984870	C₂	semidirect
3. #17984867	C₂	semidirect
4. #17984830	C₂	semidirect
5. #17984770	C₂	semidirect
6. #17984760	C₂	semidirect
7. #104×C₁₃₉	C₂	semidirect
63. C₁₃₉	#454	semidirect
70. #104	D₁₃₉	semidirect

H ≅ G/N

type

1. #17984889

C₂

semidirect

2. #17984870

C₂

semidirect

3. #17984867

C₂

semidirect

4. #17984830

C₂

semidirect

5. #17984770

C₂

semidirect

6. #17984760

C₂

semidirect

7. #104×C₁₃₉

C₂

semidirect

63. C₁₃₉

#454

semidirect

70. #104

D₁₃₉

semidirect

Subgroups — #22581127

Conjugacy classes of subgroups

Complements